The name of the group to which a molecule belongs is determined
by the symmetry elements it possesses. Grouping together molecules
with the same symmetry elements will automatically group together
molecules of the same shape eg CX_{4} where X = H, F,
Cl, Br or I; as long as the four X groups are identical, these
molecules will all fall into the same point group. The various
point groups are as follows:
C_{1}: Molecules which possess only the identity
operation. eg CBrClFI , a carbon atom with four different halogens
substituted onto it:
C_{i}: Molecules with the identity and
inversion alone. This is a relatively uncommon point group to
encounter in symmetry studies.e.g
C_{s}: Molecules with the identity and a mirror
plane alone. Again, this is not a commonly encountered point
group. e.g:
C_{n}: Molecules with the identity and a C_{n}
axis alone. Another of the rarelymet point groups. A simple
example is hydrogen peroxide, H_{2}O_{2} , which
belongs to C_{2}:
C_{nv}: Molecules with the identity,
a C_{n} axis, and n vertical mirror planes, σ_{v}.
Commonly encountered examples are water, which belongs to C_{2v}
, and ammonia, which belongs to C_{3v}. A special subset
of this group is the group C_{∞v}.
The group notation implies that all rotations around the axis
and reflections in a plane containing the axis are symmetry
operations. The group C_{∞v}
consists, therefore, of all linear molecules that do not have
a centre of inversion, for example heteronuclear diatomics and
the linear OCS molecule
C_{nh}: Molecules with the identity,
a C_{n} principle axis, and a horizontal mirror plane.
e.g. B(OH)_{3} , in the correct conformation belongs
to C_{3h}:
D_{n}: Molecules with the identity, an
nfold principle axis and n C_{2} axes perpendicular
to the C_{n} axis.
D_{nh}: Molecules with the identity,
an nfold principle axis, n perpendicular C_{2} axes,
and a horizontal mirror plane. Planar trigonal molecules
such as the BX_{3} series belong to D_{3h} ,
while benzene belongs to the group D_{6h}. There is
a subset of this group, D_{∞h}
, which contains all linear molecules which do have a centre
of inversion, for example homonuclear diatomics and ethyne (C_{2}H_{2}).
D_{nd}: Molecules with the identity,
an nfold principle axis, n C_{2} axes perpendicular
to the C_{n} axis, and n dihedral mirror planes (vertical
mirror planes which bisect the C_{2} axes). Allene,
H_{2}C=C=CH_{2} , belongs to D_{2d}:
S_{n}: Molecules that do not classify
into one of the above groups, but do possess an S_{n}
axis, belong to the group S_{n}. Molecules in this group
with n > 4 are very rare. Note there is no S_{2}
group, as the S_{2} operation is equivalent to an inversion.
Thus a molecule that would appear to belong to S_{2}
is classified as C_{i}.
Though there are examples of molecules available for all the
above groups, naturally some will be encountered more often
than others. Those likely to be encountered most often in studies
of symmetry are C_{nv} , D_{nh}
and D_{nd}. However, one should always
be alert to the existence of the other groups, and the possibility
that a molecule might classify into one of them instead.
The cubic groups, T, T_{d }, T_{h }, O,
O_{h }, I , I_{h}: These groups all have
high symmetry. They are distinguished from the above groups
by having no principle axis, but multiple equivalent C_{n}
axes with the highest number of n.
The group T_{d} is the group of the regular tetrahedron
(so would include molecules such as methane, CH_{4}).
The group T contains objects with the rotational symmetry of
a tetrahedron but none of its mirror planes, while T_{h}
is based upon the T group but contains an inversion centre.
O_{h} is the group of a regular octahedron (so would
contain molecules such as SF_{6}). O has the rotational
symmetry of the octahedron but no mirror planes. The icosahedral
group I_{h} has the symmetry of an icosahedron, and
contains the molecule C_{60}, as well as some of the
more complex hydrides of boron.
In practice, only the groups T_{d }, O_{h },
and I_{h} are likely to be encountered with any frequency.
The full rotation group, R_{3}. Contains an
infinite number of rotational axes with all possible values
of n. This is the point group of a sphere or an atom.
No molecules belong to this group.
It should be noted that the presence of certain symmetry elements
automatically indicates the presence of others. For example,
an S_{2} axis implies the presence of an inversion centre,
as the two operations are equivalent. A C_{4} axis must
also be a C_{2} axis, as the C_{2} rotation
is equivalent to two successive C_{4} rotations. Numerous
similar examples exist.
Classification of molecules into their point groups is usually
achieved with some kind of key or flow diagram which must be
followed through answering questions about the symmetry elements
an object or molecule possesses, until a classification is made.
There are various different charts available for this purpose,
an example is given on the next page.
